Gravitational potential energy formula for A-Level Physics
Gravitational potential energy (GPE) at A-Level is defined as the work done in moving a mass from infinity to a point in a gravitational field. The standard formula is E = –GMm/r, where the negative sign reflects that infinity is chosen as the zero point and gravity always pulls a mass inwards. Near the Earth's surface the simpler GCSE formula E = mgh still works, because g is roughly constant over short distances.
This guide covers why the negative sign matters, how to derive the formula from Newton's law of gravitation, and how to apply it to escape velocity, orbits, and energy-change questions in AQA Paper 2 and synoptic questions in Paper 3.
Two formulas, two contexts
E = mgh near the surface where g is constant. E = –GMm/r at A-Level distances where g changes with height.
Infinity is the zero
GPE is defined as zero at infinity and negative everywhere else. The deeper into a field, the more negative the value.
Links to escape velocity
Setting kinetic energy equal to the work needed to escape gives v = root(2GM/r). Examiners love this synoptic link.
What gravitational potential energy means at A-Level
Gravitational potential energy is the energy a mass has because of its position in a gravitational field. At A-Level the definition tightens: GPE is the work done by an external force in moving a mass from infinity to a point in the field, against gravity.
At GCSE you treated GPE as a positive quantity given by mgh. That works fine at small heights near the Earth's surface, where the gravitational field strength g is roughly 9.81 N/kg and does not change much. At A-Level you have to handle situations where g changes with distance, like satellites in orbit or rockets leaving the Earth, and the GCSE formula breaks.
Why GPE is negative The zero of GPE is chosen to be at infinity, where the gravitational field is zero. To bring a mass closer to a planet you have to release energy (gravity does positive work on it). So the energy at any finite distance is less than the energy at infinity, which means it is negative.
The A-Level formula and where it comes from
The A-Level formula for gravitational potential energy between two point masses is E = –GMm/r. Here G is the universal gravitational constant (6.67 x 10^–11 N m^2 kg^–2), M is the mass of the larger body (such as a planet), m is the mass of the smaller body, and r is the distance between their centres.
The formula comes from integrating Newton's law of gravitation. The gravitational force between two masses is F = GMm/r^2. The work done by an external force in moving the small mass from infinity to r against this force is the integral of F dr from infinity to r, which evaluates to –GMm/r. The minus sign is the result of the limits, not a mistake.
| Formula | When to use it | Notes |
|---|---|---|
| E = mgh | Near the Earth's surface, small heights | Treats g as constant. Fine for lab-bench physics |
| E = –GMm/r | Large distances, varying g | The proper A-Level form. r is measured from the centre of mass |
| ΔE = mgΔh | Energy change near the surface | Use when you only care about the change, not the absolute value |
| ΔE = GMm(1/r1 – 1/r2) | Energy change between two distances r1 and r2 | Most common form in satellite questions |
Gravitational potential vs gravitational potential energy
Gravitational potential (symbol V) and gravitational potential energy (symbol E or U) are closely linked but not the same thing. Gravitational potential is the work done per unit mass to bring a small test mass from infinity to a point in the field, measured in J/kg. Gravitational potential energy is that quantity multiplied by the actual mass m, measured in J.
The formulas mirror each other: V = –GM/r and E = mV = –GMm/r. AQA Paper 2 often asks for one and expects you to spot the link. If you remember the potential formula, just multiply by m to get the energy.
Symbols vary between exam boards AQA uses E for energy and V for potential. Some textbooks use U for energy and ϕ (phi) for potential, particularly translated European texts. Stick to your specification, but recognise the alternatives in case they appear in textbooks or past papers from other boards.
Worked example: Energy to raise a satellite
Calculate the energy needed to lift a 500 kg satellite from the Earth's surface (radius 6.37 x 10^6 m) to a geostationary orbit at 4.22 x 10^7 m from the centre of the Earth. The mass of the Earth is 5.97 x 10^24 kg.
Step 1: Calculate the GPE at the surface. E1 = –GMm/r1 = –(6.67 x 10^–11)(5.97 x 10^24)(500) / (6.37 x 10^6) = –3.13 x 10^10 J.
Step 2: Calculate the GPE at geostationary orbit. E2 = –GMm/r2 = –(6.67 x 10^–11)(5.97 x 10^24)(500) / (4.22 x 10^7) = –4.72 x 10^9 J.
Step 3: The energy required is the change. ΔE = E2 – E1 = –4.72 x 10^9 – (–3.13 x 10^10) = +2.66 x 10^10 J. Roughly 27 GJ, which is consistent with real launch energy estimates.
Escape velocity
Escape velocity is the minimum speed at which an object must be launched from a planet's surface so that it can reach infinity with zero kinetic energy. The derivation is one of the most-tested A-Level synoptic links.
At infinity, both the kinetic and potential energy are zero. At the surface, the total energy must therefore also be zero. So 1/2 mv^2 + (–GMm/r) = 0, which rearranges to v = root(2GM/r). For Earth, v = 11.2 km/s. This is independent of the mass m of the launched object, which is a satisfying result and a common follow-up question.
Escape velocity is for projectiles, not rockets The escape velocity calculation assumes the object is launched in one impulsive push, like a cannonball. A continuously thrusting rocket can leave at any speed because it is constantly adding energy. Examiners sometimes test whether students understand this distinction.
Total energy of a satellite in orbit
A satellite in a circular orbit has both kinetic and gravitational potential energy. By setting gravitational force equal to centripetal force, you find that 1/2 mv^2 = GMm/(2r), so the kinetic energy is +GMm/(2r). The gravitational potential energy is –GMm/r. Adding them gives a total energy of –GMm/(2r).
The total is negative, which makes sense: A bound orbit has less energy than a mass at infinity. To move a satellite to a higher orbit, you have to add energy. To bring it down, energy is released. This is the reasoning behind orbital manoeuvres and reentry.
| Quantity | Formula | Sign |
|---|---|---|
| Kinetic energy | +GMm / (2r) | Positive |
| Gravitational potential energy | –GMm / r | Negative |
| Total mechanical energy | –GMm / (2r) | Negative (bound orbit) |
Where students lose marks
Examiner reports from AQA flag the same handful of errors every year. They fall into three categories: Using the wrong formula, dropping the minus sign, and forgetting that r is measured from the centre of mass, not from the surface.
Mistakes that cost marks every June Using E = mgh for a satellite orbit (g is not constant up there). Dropping the negative sign and treating GPE as positive. Measuring r from the surface instead of from the centre of the planet. Forgetting to add the planet's radius to the orbital height. Using the wrong gravitational constant (G is the universal one, g is the local field strength). Mixing up potential V (J/kg) with potential energy E (J).
Worked example: Escape velocity from the Moon
Calculate the escape velocity from the Moon. The mass of the Moon is 7.35 x 10^22 kg and its radius is 1.74 x 10^6 m.
Use v = root(2GM/r). Substituting the values gives v = root(2 x 6.67 x 10^–11 x 7.35 x 10^22 / 1.74 x 10^6). The numerator inside the root is 9.80 x 10^12 and dividing by 1.74 x 10^6 gives 5.63 x 10^6. Taking the square root gives v = 2.37 km/s.
This is about a fifth of Earth's escape velocity, which is why the Apollo missions could leave the Moon with a relatively small ascent stage. The result also illustrates why the Moon has no atmosphere: Gas molecules at lunar temperatures can exceed this speed.
Key facts to memorise for AQA Paper 2
- Definition: GPE is the work done in moving a mass from infinity to a point in a gravitational field
- Formula: E = –GMm/r, where r is measured from the centre of mass
- Sign: GPE is always negative, with zero at infinity
- GCSE limit: E = mgh still works near the surface where g is approximately constant
- Gravitational potential: V = –GM/r, measured in J/kg. Multiply by m to get GPE
- Escape velocity formula is v = root(2GM/r), independent of the launched mass
- Satellite total energy in orbit: –GMm/(2r)
- Constants: G = 6.67 x 10^–11 N m^2 kg^–2, g (Earth's surface) = 9.81 N/kg
